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# Tensor Eigenvalues | Tensor 14 | Eigenvalues And Eigenvectors Of A Tensor

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## Tensor 14 | Eigenvalues And Eigenvectors Of A Tensor

Transcript:

Hello, this is a bit again, so I’m going to discuss today. All about the eigenvalues and eigenvectors of attempts of oh here is slowed down again. I can values and eigenvector’s. So what is eigenvalues? And what is eigenvectors of the tangent, so say, for example, it’s very easy, and its very important concept, especially in every effect Sciences science engineering subjects because it is really a very important topic, and it is still a very important influence on science and engineering or new medical science. Okay, so, for example, here, consider the tense of Eli and a vector, sir. I can fix over the eigenvectors, okay, So eigen vectors and eigen values are defined by the following equation. Say, say, for example. D is an eigen vector of T right, so V is an D is a vector, right, and that vector is the eigen vectors of a so we if it is eigen vectors of a then we write this in following equations, A TV equal to Lambda V right so we should be Augen vectors of density If this follows the fornication, So what is Alpha Alpha is nothing But Alpha, it’s eigen. Then I again so so V. So this kind of transformation, so heat is transferred to the tensor that transforms Ki into something that is Alpha V so here V is parallel to itself right, so we celebrate steps and alpha scalar values and its eigen value, eigen values and these eigen vectors Eigen vectors. So any tensor has infinitely many infinitely infinitely infinite number of eigen vectors, so for example. I can give you an example here. Say, for example. If you consider this year the answer say, for example, you want to T 1 + 1 T 1 1 T. And its corresponding eigen values say, for example, you is Eigen value, then 4 minus 4 right, 4 + 4 minus 4 Then you can died, according to the definition here, so we can then down here a piece of work and it’ll be 1 1 3 1 3 times 4 minus 4 so water. Can that down here? So T Times 40 L. T times T tens for here, L plus 1 times minus 4 so that is minus 4 right and again here 1 1 times 4 equal to 4 minus 2 L. So it is eight and – age right 8 and minus 8 so we can write down this the following format here we can write down this – 2 times 4 minus 4 say, see here. The 4 minus 4 is exactly same as the 4 minus 4 and 2 is the eigen values trees, an eigen value here to the in two dimensional case. I can, I can well and the eigenvector so so let’s prove this eigenvector is same as this like. Yeah, so we will go to the more details all about that characteristic polynomial equation. How can find the characteristic polynomial equation on how you can find the right in them right, so so from the definition, we know that say, for example, a Zygon, a stencil, right and V V is the eigen corresponding eigen eigen eigen vector. So you can pattern a filling the definition Lambda V so from where you can let down AV minus Lambda V equal to 0 right, yeah, so we can bring out the back end vectors from this, so we can write them a minus lambda. You bring this right so we can write down this Lambda sake, you know, just Lambda equal to Lambda. I the identity matrix. So which transforms an effect or to any vector each, that’s out there, right, So it’s same equal to the same self equal to Lambda, So we know very well that metric so can buy down here a minus. I so this eigen vectors. And these are sending the colorum so the girl for you so did the identity matrix and the eigen values and a the tensor and the eigen vector’s. Right, son, get the eigen vectors, then matrix identity tensor matrix, or a delete tensor or potential. Oliver, and this is our original tensor worried, right, so we can that on the pancakes, and so definitely Eigenvector should not be a zero vector, so this always be a nonzero vector, right, but this should be a nonzero vectors, you know, because if is it ghetto vectors then? Then, if say, for example, if it is zero vectors, then the error vector stands, any tensor would be together, so there there is many pens or something like that, so it cannot be, so this must be a nonzero vector, so it’s a linear algebraic equation, So this must be, uh. This might be there about it. Can’t be there until unless it’s determinant. But if determinant can be Cairo so determinant of determinant of a when I find tonight pigs, sorry, the water can die down in some direction from I. So what are you how I can died on this occasion here? So we can that on this? You can week and I’ve done this in here. So a say, for example here, so you get a 1 1 a 1 2 A 1 a 2 1 a 2 2 a 2 t2 on here. So, you know, so we end just to find out or to build upon, all right, so we know died in genetics here. I’d intimated 1 0 0 0 1 You know, 0 0 and want it right and Alpha Alpha corresponding eigen value to find find. All right so we can write down the pollen kitchen, then so so here a 1 1 a 1 2 A 1 a 1 a 2 2 A 2 T here is t1 it a 2 minus Lambda 1 You know, so 0 1 0 You know, wrong, so that determinant of that right so again they don’t talk. So if you, you know tight it here, so you can get them. Damned all right, Lambda here Lambda here. So you could subtract this victim to this, so we can find out that so you want to because you want to. Minus 0 is equal to a 1 to a 1 T 2 minus Lambda 2 T. It’s you on C 250 -. Lm right. The tensor and the determinant determinant our determinant of the immanent of that is equal to tell from there you can so from will have this missing from the equation for something like that. So the cactus education or polynomial equation at the steeple in the militants to put any medication, so the root of this, so as the T dimensional. Right, so we’ll get Alpha 3 or something like that, so we’ll get T bustier present block for 1 Alpha 1 Alpha 2 so these are corresponding. The again again will will get they convert something like that, right, so these this we see this all for more. This publication is called the characteristic equation. These are the colors. So I’m going to give you an example how to do that? For example, here, we have a Tensile N equal to 0 and we have to find out if Aegon been so how we can find out together, so we can dive down this determinant, a minor like that either identity matrix and for the age, I can build a well that convergence so determinant here 9 right, minus Alpha 0 here. Then this should be 3 should be 0 so I can die down, so from there, you can dive down to minus Alpha 2 2 minus Alpha is 0 and 0 0 3 minus Alpha 0 and I don’t get again 0 4 so that is equal to zero. Its determinant determinant, right, so you can get them from here that, so if you kind of don’t use a minute of that, an equal to minus Lambda. P minus Lambda Times Times Lambda minus in the fall term. It should be general, right, so from there, you can that down here. Minus Alpha Cube plus 14 Alpha Square Alpha Square minus 35 Alpha plus, so he’s a polynomial equation, right, harder, double enemy picture English, the power of our Alpha X Cube here right here. So we’ll have T rules of alpha so surprising Blanco one. So if we solve the equation, we’ll get this the roots of this year to find out the our a convinced of their here, right, so if I know to our egg invented something like and the corresponding again Bill is how we can find it, they can because but there are infinitely number of a conductors of that step, for example here, if am and Eigen Vector should be nonzero, so we already have already seen that again because mark, then on because if it is there a better, then there are better stands, and it also should be general, so it can exist, right, so say, Knowns will be any non-zero Vector say, 1 0 0 This should be can also be a eigen vectors, and this can also be regular to say 0 to minus 1 You can, and there are a lot of infinitely number of vectors or something like that, so say we can. If you want, then you can test that. What is our main deck? Just do? Yeah, but I don’t hear. Hey, hey, four zero or nine these times? Yes, this time it this time’s this so explosive that day so one, you know, Ditto our eigenvalues. Or what is the Reagan Bill? You can build up to one and eleven, right so so you’re expecting that he should be like this, so say, for example, here we can write similarly this and this. It should be all the time if we multiply this right, and if you make, we multiply this like this till we found this. So if we end up here, then you can. I can make it 1-0 engine, right, so we got this. So this is our our correspondent, so you can. In fact, we can find it so you can find out them. There are a lot of eigenvector or something that the infinite lag and Victor and eigenvector say he had a three-component academic spectacle. Miriam, better defender didn’t so it’s years. If you draw a coordinate here, then for one only day, one wanting it to the exit. So it’s quite zero and end of the eigenvector’s repeat, say, so the reaction of Eigen Eigen value eigen vector v1 is from zero to that, So it lets you know something like that. So what is the direction of that 0 to minus 1 so 0 here is a t-1000 It’s nothing but that, yeah, so it’s very simple. That is very easy. Find out that you know something like that. OK, so I hope you have got. If you’ve understood all the eigenvalues and eigenvectors on fire and so this is similar to the matrix or something at it, so I’m going to stop because of my time limitation here in the next lecture. I am going to make you understand. What is the principal values and peaceful direction of a TL symmetric tensor? So it is a really important for you class? Julie says to do research on apply center for something like facial endured an exam and you feed applying to a fault or quality on door states of strain analysis of the phone at 4 they must have to know the word of the Pittsburgh diction and the corresponding pins with them. But thank you guys, and I am going to see you on the next clip.

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